cp's OEIS Frontend

A347450 Numbers whose multiset of prime indices has alternating product <= 1.

%I A347450 #8 Sep 27 2021 07:56:09
%S A347450 1,2,4,6,8,9,10,14,15,16,18,21,22,24,25,26,32,33,34,35,36,38,39,40,46,
%T A347450 49,50,51,54,55,56,57,58,60,62,64,65,69,72,74,77,81,82,84,85,86,87,88,
%U A347450 90,91,93,94,95,96,98,100,104,106,111,115,118,119,121,122
%N A347450 Numbers whose multiset of prime indices has alternating product <= 1.
%C A347450 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A347450 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
%C A347450 Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A347450 Also numbers whose multiset of prime indices has alternating sum <= 1.
%F A347450 Union of A028982 and A119899.
%F A347450 Union of A028260 and A001105.
%e A347450 The initial terms and their prime indices:
%e A347450       1: {}            26: {1,6}           56: {1,1,1,4}
%e A347450       2: {1}           32: {1,1,1,1,1}     57: {2,8}
%e A347450       4: {1,1}         33: {2,5}           58: {1,10}
%e A347450       6: {1,2}         34: {1,7}           60: {1,1,2,3}
%e A347450       8: {1,1,1}       35: {3,4}           62: {1,11}
%e A347450       9: {2,2}         36: {1,1,2,2}       64: {1,1,1,1,1,1}
%e A347450      10: {1,3}         38: {1,8}           65: {3,6}
%e A347450      14: {1,4}         39: {2,6}           69: {2,9}
%e A347450      15: {2,3}         40: {1,1,1,3}       72: {1,1,1,2,2}
%e A347450      16: {1,1,1,1}     46: {1,9}           74: {1,12}
%e A347450      18: {1,2,2}       49: {4,4}           77: {4,5}
%e A347450      21: {2,4}         50: {1,3,3}         81: {2,2,2,2}
%e A347450      22: {1,5}         51: {2,7}           82: {1,13}
%e A347450      24: {1,1,1,2}     54: {1,2,2,2}       84: {1,1,2,4}
%e A347450      25: {3,3}         55: {3,5}           85: {3,7}
%t A347450 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A347450 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347450 Select[Range[100],altprod[primeMS[#]]<=1&]
%Y A347450 The additive version (alternating sum <= 0) is A028260.
%Y A347450 The reverse version is A028982, counted by A119620.
%Y A347450 Allowing any alternating product < 1 gives A119899.
%Y A347450 Factorizations of this type are counted by A339846, complement A339890.
%Y A347450 Allowing any alternating product >= 1 gives A344609, multiplicative A347456.
%Y A347450 Partitions of this type are counted by A347443.
%Y A347450 Allowing any integer alternating product gives A347454, reciprocal A347451.
%Y A347450 The complement is A347465, reverse A028983, counted by A347448.
%Y A347450 A056239 adds up prime indices, row sums of A112798.
%Y A347450 A236913 counts partitions of 2n with reverse-alternating sum <= 0.
%Y A347450 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A347450 A335433 lists numbers whose prime indices are separable, complement A335448.
%Y A347450 A344606 counts alternating permutations of prime indices.
%Y A347450 A347457 lists Heinz numbers of partitions with integer alternating product.
%Y A347450 Cf. A001105, A001222, A027193, A344617, A345958, A346703, A346704, A347449, A347461, A347462.
%K A347450 nonn
%O A347450 1,2
%A A347450 _Gus Wiseman_, Sep 24 2021